Abstract
In this work, we use the generalized exponential function in the fractional-order domain to define generalized cosine and sine functions. We then re-visit some important trigonometric identities and generalize them from the narrow integer-order subset to the more general fractional-order domain. It is clearly shown that trigonometric functions and trigonometric identities in the transient-time of a non-integer-order system have significantly different values from their steady-state values. Identities such as sin2(t) + cos2(t) = 1 are shown to be invalid in the transient-time of a fractional-order system. Some generalized hyperbolic functions and identities are also given in this work. Application to the evaluation of the step-response of a non-integer-order system is given.
Keywords
- Fractional-Calculus
- Generalized Exponential function
- Fractional order systems
- Generalized Trigonometric Functions
- Generalized Hyperbolic Functions
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Radwan, A.G., Elwakil, A.S. (2012). Transient-Time Fractional-Space Trigonometry and Application. In: Huang, T., Zeng, Z., Li, C., Leung, C.S. (eds) Neural Information Processing. ICONIP 2012. Lecture Notes in Computer Science, vol 7663. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34475-6_6
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DOI: https://doi.org/10.1007/978-3-642-34475-6_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34474-9
Online ISBN: 978-3-642-34475-6
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